Differential equations. An introduction to modern methods and applications.pdf

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Cover

Title Page

Contents

1 Introduction

1.1 Mathematical Models, Solutions, and Direction Fields

Terminology

Solutions and Integral Curves

Initial Value Problems

The Direction Field for u' = –k(u – T)

Solutions and Direction Fields for y' = f (t, y)

Constructing Mathematical Models

1.2 Linear Equations: Method of Integrating Factors

The Method of Integrating Factors for Solving y' + p(t)y = g(t)

1.3 Numerical Approximations: Euler’s Method

1.4 Classification of Differential Equations

CHAPTER SUMMARY

2 First Order Differential Equations

2.1 Separable Equations

2.2 Modeling with First Order Equations

2.3 Differences Between Linear and Nonlinear Equations

2.4 Autonomous Equations and Population Dynamics

2.5 Exact Equations and Integrating Factors

2.6 Accuracy of Numerical Methods

2.7 Improved Euler and Runge–Kutta Methods

CHAPTER SUMMARY

Projects

2.P.1 Harvesting a Renewable Resource

2.P.2 Designing a Drip Dispenser for a Hydrology Experiment

2.P.3 A Mathematical Model of a Groundwater Contaminant Source

2.P.4 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin

3 Systems of Two First Order Equations

3.1 Systems of Two Linear Algebraic Equations

Eigenvalues and Eigenvectors

3.2 Systems of Two First Order Linear Differential Equations

Matrix Notation, Vector Solutions, and Component Plots

Geometry of Solutions: Direction Fields and Phase Portraits

Direction Fields

Phase Portraits

General Solutions of Two First Order Linear Equations

Existence and Uniqueness of Solutions

Linear Autonomous Systems

Transformation of a Second Order Equation to a System of First Order Equations

3.3 Homogeneous Linear Systems with Constant Coefficients

On Reducing x' = Ax + b to x' = Ax

The Eigenvalue Method for Solving x' = Ax

Extension to a General System

Real and Different Eigenvalues

Nodal Sources and Nodal Sinks

Saddle Points

3.4 Complex Eigenvalues

Spiral Points and Centers

3.5 Repeated Eigenvalues

3.6 A Brief Introduction to Nonlinear Systems

3.7 Numerical Methods for Systems of First Order Equations

CHAPTER SUMMARY

Projects

3.P.1 Eigenvalue-Placement Design of a Satellite Attitude Control System

3.P.2 Estimating Rate Constants for an Open Two-Compartment Model

3.P.3 The Ray Theory of Wave Propagation

3.P.4 A Blood-Brain Pharmacokinetic Model

4 Second Order Linear Equations

4.1 Definitions and Examples

Linear Equations

Dynamical System Formulation

The Spring-Mass System

The Linearized Pendulum

The Series RLC Circuit

4.2 Theory of Second Order Linear Homogeneous Equations

Existence and Uniqueness of Solutions

Linear Operators and the Principle of Superposition for Linear Homogeneous Equations

Wronskians and Fundamental Sets of Solutions

Abel’s Equation for the Wronskian

4.3 Linear Homogeneous Equations with Constant Coefficients

The Characteristic Equation of ay" + by' + cy = 0

Initial Value Problems and Phase Portraits

4.4 Mechanical and Electrical Vibrations

Undamped Free Vibrations

Damped Free Vibrations

Phase Portraits for Harmonic Oscillators

4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

General Solution Strategy

Method of Undetermined Coefficients

Superposition Principle for Nonhomogeneous Equations

Summary: Method of Undetermined Coefficients

4.6 Forced Vibrations, Frequency Response, and Resonance

Forced Vibrations with Damping

The Frequency Response Function

Forced Vibrations without Damping

4.7 Variation of Parameters

Variation of Parameters for Linear Second Order Equations

CHAPTER SUMMARY

Projects

4.P.1 A Vibration Insulation Problem

4.P.2 Linearization of a Nonlinear Mechanical System

4.P.3 A Spring-Mass Event Problem

4.P.4 Uniformly Distributing Points on a Sphere

4.P.5 Euler–Lagrange Equations

5 The Laplace Transform

5.1 Definition of the Laplace Transform

Existence of the Laplace Transform

5.2 Properties of the Laplace Transform

5.3 The Inverse Laplace Transform

Partial Fractions

5.4 Solving Differential Equations with Laplace Transforms

Characteristic Polynomials and Laplace Transforms of Differential Equations

5.5 Discontinuous Functions and Periodic Functions

5.6 Differential Equations with Discontinuous Forcing Functions

5.7 Impulse Functions

5.8 Convolution Integrals and Their Applications

5.9 Linear Systems and Feedback Control

CHAPTER SUMMARY

Projects

5.P.1 An Electric Circuit Problem

5.P.2 Effects of Pole Locations on Step Responses of Second Order Systems

5.P.3 The Watt Governor, Feedback Control, and Stability

6 Systems of First Order Linear Equations

6.1 Definitions and Examples

First Order Linear Systems: General Framework

Applications Modeled by First Order Linear Systems

6.2 Basic Theory of First Order Linear Systems

6.3 Homogeneous Linear Systems with Constant Coefficients

The Matrix A Is Nondefective With Real Eigenvalues

6.4 Nondefective Matrices with Complex Eigenvalues

6.5 Fundamental Matrices and the Exponential of a Matrix

Fundamental Matrices

The Matrix Exponential Function e[sup(At)]

Methods for Constructing e[sup(At)]

6.6 Nonhomogeneous Linear Systems

Variation of Parameters

Undetermined Coefficients and Frequency Response

6.7 Defective Matrices

CHAPTER SUMMARY

Projects

6.P.1 A Compartment Model of Heat Flow in a Rod

6.P.2 Earthquakes and Tall Buildings

6.P.3 Controlling a Spring-Mass System to Equilibrium

7 Nonlinear Differential Equations and Stability

7.1 Autonomous Systems and Stability

7.2 Almost Linear Systems

7.3 Competing Species

7.4 Predator–Prey Equations

7.5 Periodic Solutions and Limit Cycles

7.6 Chaos and Strange Attractors: The Lorenz Equations

CHAPTER SUMMARY

Projects

7.P.1 Modeling of Epidemics

7.P.2 Harvesting in a Competitive Environment

7.P.3 The Rossler System

A: Matrices and Linear Algebra

A.1 Matrices

A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank

A.3 Determinants and Inverses

A.4 The Eigenvalue Problem

B: Complex Variables

ANSWERS TO SELECTED PROBLEMS

REFERENCES

PHOTO CREDITS

INDEX

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P1: OSO/OVY JWCL373-FM P2: OSO/OVY JWCL373-Brannan-v1 QC: OSO/OVY T1: OSO October 30, 2010 18:19 ALL THE HELP, RESOURCES, AND PERSONAL SUPPORT YOU AND YOUR STUDENTS NEED! www.wileyplus.com/resources 2-Minute Tutorials and all of the resources you & your students need to get started. Student support from an experienced student user. Collaborate with your colleagues, find a mentor, attend virtual and live events, and view resources. www.WhereFacultyConnect.com Pre-loaded, ready-to-use assignments and presentations. Created by subject matter experts. Technical Support 24/7 Your WileyPLUS Account Manager. FAQs, online chat, and phone support. www.wileyplus.com/support Personal training and implementation support. ii CONFIRMING PAGES

P1: OSO/OVY JWCL373-FM P2: OSO/OVY JWCL373-Brannan-v1 QC: OSO/OVY T1: OSO October 30, 2010 18:19 DIFFERENTIAL EQUATIONS An Introduction to Modern Methods and Applications iii CONFIRMING PAGES

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P1: OSO/OVY JWCL373-FM P2: OSO/OVY JWCL373-Brannan-v1 QC: OSO/OVY T1: OSO October 30, 2010 18:19 DIFFERENTIAL EQUATIONS An Introduction to Modern Methods and Applications 2nd Edition James R. Brannan Clemson University William E. Boyce Rensselaer Polytechnic Institute John Wiley & Sons, Inc. v CONFIRMING PAGES

P1: OSO/OVY JWCL373-FM P2: OSO/OVY JWCL373-Brannan-v1 QC: OSO/OVY T1: OSO October 30, 2010 18:19 PUBLISHER ACQUISITIONS EDITOR ASSISTANT EDITOR SENIOR EDITORIAL ASSISTANT MARKETING MANAGER CREATIVE DIRECTOR COVER DESIGN COVER PHOTO SENIOR PRODUCTION EDITOR SENIOR PHOTO EDITOR Laurie Rosatone David Dietz Jeffrey Benson Pamela Lashbrook Sarah Davis Harry Nolan Diane Gerard Jorgen Schytte/Photolibrary Sujin Hong Mary Ann Price This book was set in 10/12 Times by Aptara®, Inc. and printed and bound by Donnelley/Von Hoffmann. The cover was printed by Donnelley/Von Hoffmann. This book is printed on acid free paper. ∞ Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulﬁll their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper speciﬁcations and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship. Copyright c⃝ 2007, 2011 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748- 6008, website www.wiley.com/go/permissions. Evaluation copies are provided to qualiﬁed academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licenced and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. ISBN 978-0-470-45824-2 ISBN 978-1-118-01187-4 (BRV) Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 vi CONFIRMING PAGES

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